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LVI OM - I - Problem 12
Given are functions
(7-fold iteration of function $ f $). Determine whether the number $ g(3)-g(0) $ is divisible by the number $ g(2)-g(0) $. | The numbers $ g(0) $, $ g(2) $, and $ g(3) $ are the terms $ a_7 $, $ b_7 $, and $ c_7 $ of the sequences $ (a_n) $, $ (b_n) $, and $ (c_n) $ defined recursively by the formulas
We will show that the number $ c_7 - a_7 = g(3) - g(0) $ is not divisible by $ b_7 - a_7 = g(2) - g(0) $. In the proof, we will use the follo... | proof | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XL OM - I - Task 5
For a given natural number $ n $, determine the number of sequences $ (a_1, a_2, \ldots , a_n) $ whose terms $ a_i $ belong to the set $ \{0,1,2,3,4\} $ and satisfy the condition $ |a_i - a_{i+1}| = 1 $ for $ i = 1,2,\ldots,n-1 $. | We will divide all sequences satisfying the conditions given in the problem into three types:
sequences of type I — ending with the symbol $ 0 $ or $ 4 $;\\
sequences of type II — ending with the symbol $ 1 $ or $ 3 $;\\
sequences of type III — ending with the symbol $ 2 $.
Let $ x_n $, $ y_n $, $ z_n $ denote t... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
LVII OM - I - Problem 4
Participants in a mathematics competition solved six problems, each graded with one of the scores 6, 5, 2, 0. It turned out that
for every pair of participants $ A, B $, there are two problems such that in each of them $ A $ received a different score than $ B $.
Determine the maximum number ... | We will show that the largest number of participants for which such a situation is possible is 1024. We will continue to assume that the permissible ratings are the numbers 0, 1, 2, 3 (instead of 5 points, we give 4, and then divide each rating by 2).
Let $ P = \{0,1,2,3\} $ and consider the set
Set $ X $ obviou... | 1024 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XV OM - III - Problem 5
Given an acute angle and a circle located inside this angle. Determine a point $ M $ on the given circle so that the sum of the distances from point $ M $ to the sides of the angle is minimized. | Let \( \measuredangle AOB = \delta \) (Fig. 19) be a given angle, \( M \) an arbitrary point on a given circle, and \( MN = d_1 \) and \( MP = d_2 \) the distances from point \( M \) to the rays \( OA \) and \( OB \). Let \( MO = d \), \( \measuredangle MOA = \alpha \), \( \measuredangle MOB = \beta \); assume that \( ... | notfound | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
XXVI - I - Problem 12
In space, there is a cube with side $ a $, and spheres $ B_1, B_2, \ldots, B_n $ of arbitrary radii such that every point of the cube belongs to at least one of the spheres. Prove that among these spheres, one can select pairwise disjoint spheres such that the sum of their volumes is not less tha... | By induction on $ N $ we will prove
Theorem. If in a space there is a set $ F $ of volume $ V $ contained in the sum of $ N $ open balls $ B_1, B_2, \ldots, B_N $, then there exists a subset $ B_{i_1}, B_{i_2}, \ldots, B_{i_r} $ of these balls such that the balls belonging to this subset are pairwise disjoint and the s... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
XIII OM - I - Problem 10
Prove that if
and $ A $, $ B $, $ C $ are acute angles, then
Note: The mathematical expressions and symbols are kept as they are, since they are universal and do not require translation. | To prove that the sum of the acute angles $A$, $B$, $C$ is less than $180^\circ$, it suffices to prove that $\cos \frac{A+B+C}{2} > 0$.
Notice that each of the angles $\frac{A-B+C}{2}$, $\frac{A+B-C}{2}$, $\frac{B+C-A}{2}$ lies between $-45^\circ$ and $90^\circ$, so the cosines of these angles are positive. Inequality ... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
LIX OM - I - Task 2
Given is a convex angle with vertex $ P $ and a point $ A $ lying inside this angle. Points $ X $ and $ Y $ lie on different
sides of this angle, such that $ PX = PY $ and the value of the sum $ AX + AY $ is the smallest. Prove that
$ \measuredangle XAP = \measuredangle YAP $. | Let $ B $ be a point in the plane such that $ PB = PA $ and
$ \measuredangle BPA = \measuredangle YPX $ (Fig. 1).
om59_1r_img_1.jpg
We then have $ \measuredangle BPY = \measuredangle APX $, so triangles $ BPY $ and $ APX $ are congruent
(side-angle-side criterion). From this, we obtain $ BY = AX $. The value of the ex... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XLIV OM - II - Problem 4
Let $ (x_n) $ be a sequence of natural numbers such that:
Prove that for every natural number $ k $ there exist such numbers $ r $ and $ s $ that $ x_r - x_s = k $. | Let's establish a natural number $ k $. To find the numbers $ r $ and $ s $ in question, it suffices to consider the initial segment of the sequence $ (x_n) $, consisting of its first $ k +1 $ terms:
We partition the set of natural numbers from $ 1 $ to $ 2k $ into the sum of $ k $ two-element sets, as follows:
... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XXXII - I - Problem 4
On the sides of an acute triangle $ ABC $, squares $ ABED $, $ BCGF $, $ ACHI $ are constructed outside the triangle. Prove that the medians of the triangle formed by the lines $ EF $, $ GH $, $ DI $ are perpendicular to the sides of triangle $ ABC $. | Let $K$, $L$, $M$ be the vertices of the formed triangle lying at the squares $ABED$, $BCGF$, $ACHI$, respectively. Let $KO$ and $LO$ be lines perpendicular to segments $\overline{AB}$ and $\overline{BC}$, respectively (Fig. 8).
om32_1r_img_8.jpg
Triangles $KLO$ and $BEF$ have corresponding sides that are parallel, so ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LVIII OM - III - Problem 1
In an acute triangle $ABC$, point $O$ is the center of the circumscribed circle, segment $CD$ is the altitude, point $E$ lies on side $AB$, and point $M$ is the midpoint of segment $CE$. The line perpendicular to line $OM$ and passing through point $M$ intersects lines $AC$, $BC$ at points $... | Let's draw a line through point $ M $ parallel to side $ AB $, which intersects segments $ AC $ and $ BC $ at points $ P $ and $ Q $ respectively (Fig. 12). Then points $ P $ and $ Q $ are the midpoints of sides $ AC $ and $ BC $ respectively. Since point $ O $ is the center of the circumcircle of triangle $ ABC $, we ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXI - III - Task 3
We choose an integer $ k $ from the interval $ [1, 99] $. Then we toss a coin 100 times (the outcomes of the tosses are independent). Let
Denote by $ M_k $ the event that there exists a number $ i $ such that $ k + \varepsilon_1 + \ldots + \varepsilon_i = 100 $. What should $ k $ be so that ... | Let $ P(M_k) $ be the probability of the event $ M_k $. Of course, $ P(M_{99})=\frac{1}{2} $, $ p(M_{98})=\frac{1}{2} + \frac{1}{2} \cdot \frac{1}{2} = \frac{3}{4} $. We will show that for $ k < 98 $, $ P(M_k) < \frac{3}{4} $. First, notice that
Indeed, if a head (probability of this is $ \frac{1}{2} $) comes up on th... | 98 | Combinatorics | math-word-problem | Incomplete | Yes | olympiads | false |
IV OM - II - Task 2
Prove that the equation
in which $ a < b < c < d $, has two real roots. | Let's consider the left side of equation (1), i.e., the quadratic function
We know from algebra that a quadratic function has two real roots (or zeros) if and only if it takes on both positive and negative values. If $ x $ is greater than each of the numbers $ a $, $ b $, $ c $, $ d $, then each of the differences $ x... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
II OM - I - Task 3
In a circle, two equal chords $ AB $ and $ AC $ and an arbitrary chord $ AD $ are drawn; the line $ AD $ intersects the line $ BC $ at point $ E $. Prove that the product $ AE \cdot AD $ does not depend on the position of point $ D $ on the circle, i.e., that $ AE \cdot AD = AC^2 $. | When writing the desired equality in the form of a proportion
we observe that this equality expresses the proportionality of two pairs of sides of triangles $AEC$ and $ACD$ with a common angle $A$.
The proof thus reduces to showing that triangles $AEC$ and $ACD$ are similar.
It suffices for this purpose to indicate an... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXIII OM - I - Problem 7
A broken line contained in a square with a side length of 50 has the property that the distance from any point of this square to it is less than 1. Prove that the length of this broken line is greater than 1248. | Let the broken line $ A_1A_2 \ldots A_n $ have the property given in the problem. Denote by $ K_i $ ($ i= 1, 2, \ldots, n $) the circle with center at point $ A_i $ and radius of length $ 1 $, and by $ F_i $ ($ i= 1, 2, \ldots, n-1 $) the figure bounded by segments parallel to segment $ \overline{A_iA_{i+1}} $ and at a... | 1248 | Geometry | proof | Yes | Yes | olympiads | false |
XXV OM - II - Problem 6
Given is a sequence of integers $ a_1, a_2, \ldots, a_{2n+1} $ with the following property: after discarding any term, the remaining terms can be divided into two groups of $ n $ terms each, such that the sum of the terms in the first group is equal to the sum of the terms in the second. Prove ... | \spos{1} First, note that if a sequence of real numbers satisfies the conditions of the problem, then so does any sequence of the form
where $ k $ is any real number.
Suppose that a sequence (1) of integers satisfies the conditions of the problem and $ a_1 \ne a_2 $. Without loss of generality, we can assu... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XXXVIII OM - III - Problem 3
Given is a polynomial $ W $ with non-negative integer coefficients. We define a sequence of numbers $ (p_n) $, where $ p_n $ is the sum of the digits of the number $ W(n) $. Prove that some number appears in the sequence $ (p_n) $ infinitely many times. | Let $ W(x) = a_mx^m + \ldots +a_1x+a_0 $ ($ a_i $ - non-negative integers) and let $ q $ be the number of digits in the decimal representation of the largest of the numbers $ a_0, \ldots, a_m $. Let $ \overline{a_i} $ denote the $ q $-digit decimal representation of $ a_i $; if $ a_i $ has fewer than $ q $ digits, we p... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XX OM - II - Task 5
Prove that if, during a parallel projection of one plane onto another plane, the image of a certain square is a square, then the image of any figure is a figure congruent to it. | Let the parallel projection of square $MNPQ$ lying on plane $\alpha$ be square $M'$ on plane $\beta$.
Then the projection of plane $\alpha$ is the entire plane $\beta$, and the projection is a one-to-one mapping from one plane to the other. It is known that under such a projection, the projections of parallel segments ... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XXI OM - I - Problem 2
Given is the sequence $ \{c_n\} $ defined by the formulas $ c_1=\frac{a}{2} $, $ c_{n+1}=\frac{a+c_n^2}{2} $ where $ a $ is a given number satisfying the inequality $ 0 < a < 1 $. Prove that for every $ n $ the inequality $ c_n < 1 - \sqrt{1 - a} $ holds. Prove the convergence of the sequence $ ... | From the definition of the sequence $ \{c_n\} $, it follows that all its terms are positive. By applying the method of induction, we will show that for every natural number $ n $, the inequality
holds.
We first state that $ c_1 < 1 - \sqrt{1 - a} $, i.e.,
Specifically, we have
This implies (2).
If for some natural ... | 1-\sqrt{1-} | Algebra | proof | Yes | Yes | olympiads | false |
XLIX OM - II - Problem 4
Find all pairs of integers $ x $, $ y $ satisfying the equation | Let's assume that the integers $ x $, $ y $ satisfy the given equation and that $ y \neq 0 $; then also $ x \neq 0 $. The number $ x $ is one of the roots of the quadratic trinomial $ t^2 - 1998t + 3y^2 $; let $ z $ be the other root. From the formulas $ x + z = 1998 $, $ xz = 3y^2 $, it follows that $ z $ is also an i... | (0,0),(1998,0),(648,540),(648,-540),(1350,540),(1350,-540) | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
XLIX OM - I - Problem 11
In a tennis tournament, $ n $ players participated. Each played one match against each other; there were no draws. Prove that there exists a player $ A $ who has either directly or indirectly defeated every other player $ B $, i.e., player $ A $ won against $ B $ or player $ A $ defeated some ... | Let $ A $ be the participant in the tournament who defeated the largest number of opponents. (If several players have the same maximum number of victories, we choose any one of them.) We claim that the player $ A $ selected in this way has defeated, directly or indirectly, all other players.
Let $ B $ be any player who... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
III OM - I - Problem 10
Prove that in any triangle, the radius $ R $ of the circumscribed circle and the radius $ r $ of the inscribed circle satisfy the inequality
The translation is provided as requested, maintaining the original text's line breaks and format. | Let triangle $ABC$ be inscribed in circle $O(R)$ (the symbol $O(R)$ denotes a circle with center $O$ and radius $R$) and circumscribed around circle $S(r)$ (Fig. 21). Suppose that $AC \ne BC$ and consider the isosceles triangle $ABC_1$ inscribed in the same circle $O(R)$, with point $C_1$ lying on the same side of line... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
II OM - III - Task 1
A beam of length $ a $ has been suspended horizontally by its ends on two parallel ropes of equal length $ b $. We rotate the beam by an angle $ \varphi $ around a vertical axis passing through the center of the beam. By how much will the beam be raised? | When solving geometric problems, a properly executed drawing is an important aid to our imagination. We represent spatial figures through mappings onto the drawing plane. There are various ways of such mapping. In elementary geometry, we most often draw an oblique parallel projection of the figure; in many cases, it is... | \sqrt{b^2-(\frac{} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXX OM - I - Task 9
Let $ r_k(n) $ be the remainder of the division of the natural number $ n $ by the natural number $ k $ and
Prove that for every natural number $ m $ the equality $ R(2^m-1) = R(2^m) $ holds. | From the definition of the number $ r_k (n) $, it follows that for $ n \geq 2 $ and $ k \geq 1 $, the following relationships hold:
If $ k \mid n $, then $ r_k(n) = 0 $ and $ r_k(n - 1) = k-1 $,
If $ k \mid n $, then $ r_k(n) = r_k(n - 1) + 1 $.
We also have
Therefore, for $ n \geq 2 $, we have
wh... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XIV OM - III - Task 1
Prove that two natural numbers, whose digits are all ones, are relatively prime if and only if the numbers of their digits are relatively prime. | Let $ J_m $ denote the $ m $-digit number whose digits are all ones:
If $ m $ is divisible by $ d $ ($ m $, $ d $ - natural numbers), then $ J_m $ is divisible by $ J_d $. Indeed, if $ m = k \cdot d $ ($ k $ - a natural number), then
where the number $ M $ is an integer.
$ \beta $) If $ m > n $, then
Let the numbers... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
LX OM - III - Task 1
Each vertex of a convex hexagon is the center of a circle with a radius equal to the length of the longer of the two sides of the hexagon containing that vertex. Prove that if the intersection of all six circles (considered with their boundaries) is non-empty, then the hexagon is regular. | Let's denote the consecutive vertices of the given hexagon by the letters $A, B, C, D, E$, and $F$.
Assume that the intersection of the six circles mentioned in the problem is non-empty and that point
$P$ belongs to this intersection. Consider any side of the hexagon, for the sake of argument, let it be
side $AB$. From... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LV OM - I - Task 8
Point $ P $ lies inside the tetrahedron $ ABCD $. Prove that | Let $ K $ be the point of intersection of the plane $ CDP $ with the edge $ AB $, and let $ L $ be the point of intersection of the plane $ ABP $ with the edge $ CD $ (Fig. 4).
Similarly, let $ M $ be the point of intersection of the plane $ ADP $ and the edge $ BC $, and let $ N $ denote the common point of the plane ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LVII OM - III - Problem 2
Determine all positive integers $ k $ for which the number $ 3^k+5^k $ is a power of an integer with an exponent greater than 1. | If $ k $ is an even number, then the numbers $ 3^k $ and $ 5^k $ are squares of odd numbers, giving a remainder of 1 when divided by 4. Hence, the number $ 3^k + 5^k $ gives a remainder of 2 when divided by 4, and thus is divisible by 2 but not by $ 2^2 $. Such a number cannot be a power of an integer with an exponent ... | 1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
L OM - I - Task 4
Given real numbers $ x $, $ y $, such that the numbers $ x + y $, $ x^2 + y^2 $, $ x^3 + y^3 $, and $ x^4 + y^4 $ are integers. Prove that for every positive integer $ n $, the number $ x^n + y^n $ is an integer. | Since $ 2xy= (x + y)^2 - (x^2 + y^2) $ and $ 2x^2y^2 = (x^2 + y^2)^2 - (x^4 + y^4) $, the numbers $ 2xy $ and $ 2x^2y^2 $ are integers. If the number $ xy $ were not an integer, then $ 2xy $ would be odd. But then the number $ 2x^2y^2 = (2xy)^2/2 $ would not be an integer. The conclusion is that the number $ xy $ is an... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XXXIII OM - II - Task 1
Prove that if $ c, d $ are integers, where $ c \neq d $, $ d > 0 $, then the equation
has no more than one rational root. | We will apply a known theorem:
If a rational number written in the form of an irreducible fraction $ \frac{p}{q} $ is a root of the equation $ a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0 $ with integer coefficients, then the number $ p $ is a divisor of the coefficient $ a_0 $, and the number $ q $ is a divisor of $ a_n... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XLVIII OM - III - Problem 1
Positive integers $ x_1 $, $ x_2 $, $ x_3 $, $ x_4 $, $ x_5 $, $ x_6 $, $ x_7 $ satisfy the conditions:
Determine $ x_7 $. | Let's denote: $ a = x_1+x_2 $, $ b = x_3 $, $ c = x_2+x_3 $. Then $ x_2 = c - b $, $ x_1 = a-x_2=a+b-c $. These are positive numbers by assumption. Thus
Further, $ x_4 = ab, x_5 = abc, x_6 = abc(ab + b) = a(a + 1)b^2c $. From inequality (1), it follows that $ a > 1 $. The only divisors of the number $ 144 $ that ha... | 3456 | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
V OM - III - Task 5
Prove that if in a tetrahedron $ABCD$ the opposite edges are equal, i.e., $AB = CD$, $AC = BD$, $AD = BC$, then the lines passing through the midpoints of opposite edges are mutually perpendicular and are axes of symmetry of the tetrahedron. | Let $ K $, $ L $, $ M $, $ N $, $ P $, $ Q $ denote the midpoints of the edges of the tetrahedron $ ABCD $, as indicated in Fig. 42. It is sufficient to prove that any of the lines $ KL $, $ MN $, $ PQ $, for example, $ KL $, is an axis of symmetry of the tetrahedron and that it is perpendicular to one of the two remai... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXI OM - III - Task 3
Prove that $ n > 1 $ is a prime number if and only if for every natural $ k $, such that $ 1 \leq k \leq n - 1 $, the binomial coefficient
is divisible by $ n $. | If $ n $ is a prime number, then for $ 1 \leq k \leq n - 1 $, the numbers $ k $ and $ n - k $ are less than $ n $, and therefore the numbers $ k! $ and $ (n - k)! $ are not divisible by $ n $.
From the formula $ \displaystyle \binom{n}{k} = \frac{n!}{k!(n-k)!} $ we get
Since the left side (1) is divisible by the prime... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XXII OM - III - Zadanie 1
Dowieść, że jeżeli $ a_n $ jest ciągiem różnych liczb naturalnych, których rozwinięcia dziesiętne nie zawierają, cyfry O, to
|
Oznaczmy przez $ b_n $ liczbę powstającą z liczby $ a_n $ przez zastąpienie wszystkich jej cyfr prócz pierwszej zerami. Mamy oczywiście $ b_n \leq a_n $ i wobec tego $ \frac{1}{a_n} \leq \frac{1}{b_n} $. W ciągu $ \{a_n\} $ jest $ 9 $ liczb jednocyfrowych, $ 9^2 $ liczb dwucyfrowych i ogólnie $ 9^k $ liczb $ k $-cyfr... | proof | Number Theory | proof | Incomplete | Incomplete | olympiads | false |
XL OM - III - Task 6
Prove that for any positive numbers $ a, b, c, d $ the following inequality is satisfied
Note: The original text did not provide the specific inequality to be proven. If you have the specific inequality, please provide it, and I can help translate that as well. | Let's denote:
The numbers $ A $, $ B $, $ C $, $ D $ are the coefficients of the polynomial
This is a polynomial of the fourth degree, having four negative roots: $ -a $, $ -b $, $ -c $, $ -d $ (if any of these numbers are equal, we have a multiple root).
We differentiate the polynomial $ P $:
By Rol... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
LI OM - III - Task 2
Given is a triangle $ ABC $, where $ AC = BC $. Point $ P $ lies inside triangle $ ABC $, such that $ \measuredangle PAB = \measuredangle PBC $. Point $ M $ is the midpoint of side $ AB $. Prove that | First, assume that point $ P $ lies inside triangle $ AMC $ or on segment $ MC $.
om51_3r_img_1.jpg
om51_3r_img_2.jpg
Let $ Q \neq C $ be the point of intersection of the circumcircle of triangle $ BCP $ with line $ CM $ (Fig. 1 and 2). Since $ \measuredangle PAM = \measuredangle PBC = \measuredangle PQC $, points $ A ... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXXVIII OM - I - Problem 5
Prove that if the function
has five zeros in the interval $ (0; 2\pi) $, then $ f(x) = 0 $ for every $ x $. | According to the assumption, there exist numbers $ x_i \in (0; 2\pi) $, $ i = 1, 2, 3, 4, 5 $, different from each other, such that $ f(x_i) = 0 $.
We apply the substitution $ t = \ctg(x/2) $. As $ x $ runs through the interval $ (0; 2\pi) $, $ t $ runs through the interval $ (-\infty; \infty) $, with different values ... | proof | Calculus | proof | Incomplete | Yes | olympiads | false |
XXIV OM - I - Problem 12
In a class of n students, a Secret Santa event was organized. Each student draws the name of the person for whom they are to buy a gift, so student $ A_1 $ buys a gift for student $ A_2 $, $ A_2 $ buys a gift for $ A_3 $, ..., $ A_k $ buys a gift for $ A_1 $, where $ 1 \leq k \leq n $. Assumin... | In the result of any drawing, each student draws a certain student from the same class (possibly themselves), and different students draw different students. Therefore, the result of each drawing defines a certain one-to-one mapping (permutation) of the set of all students in the class onto itself. Conversely, each per... | \frac{1}{n} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XXXV OM - I - Problem 7
Prove that for any natural numbers $ m, p $ greater than 1, the inequality is satisfied
Note: The original text was in Polish, and the problem statement is about proving an inequality for natural numbers greater than 1. However, the specific inequality to be proven is not provided in ... | We estimate the $ k $-th term of the considered sum:
since the sum appearing before the inequality sign has $ p $ terms, each of which is less than $ 1 $. Therefore,
| proof | Inequalities | proof | Incomplete | Incomplete | olympiads | false |
VIII OM - I - Task 6
Find a four-digit number, whose first two digits are the same, the last two digits are the same, and which is a square of an integer. | If $ x $ is the number sought, then
where $ a $ and $ b $ are integers satisfying the inequalities $ 0 < a \leq 9 $, $ 0 \leq b \leq 9 $. The number $ x $ is divisible by $ 11 $, since
Since $ x $ is a perfect square, being divisible by $ 11 $ it must be divisible by $ 11^2 $, so the number
is divisib... | 7744 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XXV - I - Problem 3
Prove that the bisectors of the angles formed by the lines containing the opposite sides of a convex quadrilateral inscribed in a circle are respectively parallel to the bisectors of the angles formed by the lines containing the diagonals of this quadrilateral. | Let $ A $, $ B $, $ C $, $ D $ be consecutive vertices of a quadrilateral inscribed in a circle. Let $ O $ be the point of intersection of the lines $ AB $ and $ CD $, and let $ k $ be the angle bisector of $ \measuredangle AOD $ (Fig. 12). If $ A' $ and $ C' $ are the images of points $ A $ and $ C $, respectively, un... | proof | Geometry | proof | Yes | Yes | olympiads | false |
VI OM - I - Problem 11
Prove that in an acute triangle, the sum of the distances from the orthocenter (i.e., the point of intersection of the altitudes) to the vertices is twice the sum of the radii of the circumscribed and inscribed circles. | The proof of the theorem can be conducted without using trigonometric relationships. Let's draw through each vertex of the triangle a line parallel to the opposite side and denote the points of intersection of these lines with the letters $A$, $B$, $C$ as indicated in Fig. 7. Triangle $A$ is similar to triangle $ABC$ i... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XIX OM - I - Problem 5
Given a set of $ 2n $ ($ n \geq 2 $) different numbers. How to divide this set into pairs so that the sum of the products of the numbers in each pair is a) the smallest, b) the largest? | By a partition of a set of numbers $ Z $ into pairs, we mean a set of pairs of numbers from $ Z $ such that each number in $ Z $ belongs to one and only one pair.
Let us denote the given numbers by the letters $ a_1, a_2, \ldots, a_{2n} $ in such a way that
The number of all partitions of the set of numbers (1) in... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
VII OM - I - Task 2
Prove that if $ a $ and $ b $ are integers, then the number
is divisible by $ 30 $. | It is enough to prove that $N$ is divisible by each of the numbers $2$, $3$, $5$.
a) $N$ is divisible by $2$, because if neither of the numbers $a$ and $b$ is divisible by $2$, then $a^2$ and $b^2$ are odd numbers, and thus $a^2 - b^2$ is divisible by $2$.
b) To prove the divisibility of $N$ by $3$, note first that the... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XXXIV OM - I - Problem 1
$ A $ tosses a coin $ n $ times, $ B $ tosses it $ n+1 $ times. What is the probability that $ B $ will get more heads than $ A $? | Consider the following events. $ Z_0 $ - the event that player $ B $ has flipped more heads than player $ A $, $ Z_r $ - the event that $ B $ has flipped more tails than player $ A $. These events are equally probable: if the coin is fair, it does not matter which side is called heads and which is called tails. Thus, $... | \frac{1}{2} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XLVII OM - I - Problem 4
A line tangent to the circle inscribed in an equilateral triangle $ ABC $ intersects the sides $ AB $ and $ AC $ at points $ D $ and $ E $, respectively. Prove that
$ \frac{|AD|}{|DB|} + \frac{|AE|}{|EC|} = 1 $. | Note: One of the simple tangents to the circle inscribed in triangle $ABC$ is the line $BC$. For this line, point $D$ coincides with $B$, point $E$ coincides with $C$, and the considered fractions lose their meaning. The failure to exclude such a situation in the problem statement was an obvious oversight$ \ldots $ In ... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXXVI OM - III - Problem 6
Prove that if in a convex polyhedron with $ k $ faces there are more than $ k/2 $ faces, no two of which have a common edge, then it is impossible to inscribe a sphere in this polyhedron. | Suppose that a sphere can be inscribed in a given polyhedron. Let $S_1, \ldots, S_k$ be the faces of the polyhedron, and $P_1, \ldots, P_k$ the points of tangency of the respective faces with the sphere. If $A$ and $B$ are common vertices of the faces $S_i$ and $S_j$, then the triangles $ABP_i$ and $ABP_j$ are congruen... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXIII OM - III - Problem 5
Prove that all subsets of a finite set can be arranged in a sequence, where consecutive terms differ by one element. | Assume that a given finite set $A$ has $n$ elements. By applying induction on $n$, we will prove that there exists such a sequence of all subsets of set $A$ that
(1)
all subsets of set $A$ are included, and
(2)
for any two consecutive subsets $A_i$ and $A_{i+1}$ in the sequence, one of the sets $A_{i+1} - A_i$ an... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
X OM - III - Task 4
Prove that if a quadratic equation
with integer coefficients has a rational root, then at least one of the numbers $ a $, $ b $, $ c $ is even. | \spos{1} Suppose that the number $ \frac{p}{q} $, where $ p $ and $ q $ are integers that are relatively prime, is a root of equation (1). After substituting $ x = \frac{p}{q} $ into equation (1) and multiplying by $ q^2 $, we obtain
Notice that the sum of the integers $ ap^2 $, $ bpq $, and $ cq^2 $ can only be zero ... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XI OM - II - Task 3
Given are two circles with a common center $ O $ and a point $ A $. Construct a circle with center $ A $ intersecting the given circles at points $ M $ and $ N $, so that the line $ MN $ passes through the point $ O $. | Suppose a circle with center $A$ intersects the given circles at points $M$ and $N$ lying on a straight line with point $O$. Let $S$ be the midpoint of segment $MN$. The problem will be solved when we can find point $S$. If points $M$ and $N$ lie on the same side of point $O$, then
if, however, $M$ and $N$ lie on oppo... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
LIV OM - II - Task 5
Point $ A $ lies outside the circle $ o $ with center $ O $. From point $ A $, two tangent lines to circle $ o $ are drawn at points $ B $ and $ C $. A certain tangent to circle $ o $ intersects segments $ AB $ and $ AC $ at points $ E $ and $ F $, respectively. Lines $ OE $ and $ OF $ intersect s... | Let $ X $ be the point of tangency of circle $ o $ with line $ EF $ (Fig. 3). From the equality
of
it follows that triangles $ BEP $ and $ XEP $ are congruent. Therefore, $ BP = XP $.
Similarly, we prove the equality $ CQ = XQ $. This means that triangle $ PQX $ is constructed from segments $ BP $, $ PQ $, and $ QC $... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LII OM - I - Problem 12
We consider sequences of integers $ x_0,x_1,\ldots,x_{2000} $ satisfying the conditions
Find the smallest value of the expression $ |x_1 + x_2 +\ldots + x_{2000}| $. | Answer: The smallest value of the given expression is $ 12 $.
We will prove by induction that for any sequence $ (x_n) $ satisfying the conditions of the problem, the following equality holds
For $ n = 0 $, both sides of the above equality are equal to $ 0 $. So, assume that the relation (1) is true for some $ n... | 12 | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
XXXVII OM - III - Problem 3
Prove that if $ p $ is a prime number, and the integer $ m $ satisfies the inequality $ 0 \leq m < p—1 $, then $ p $ divides the number $ \sum_{j=1}^p j^m $. | Let's denote the sum $1^m + 2^m + \ldots + p^m$ by $A(p, m)$. We need to prove that under the given conditions
Proof is conducted by induction on $m$ (for a fixed $p$). When $m = 0$, we get the number $A(p, 0) = p$, which is clearly divisible by $p$.
Fix an integer $m$ satisfying $0 < m < p - 1$ and assume that
We wa... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XLVI OM - II - Zadanie 1
Wielomian $ P(x) $ ma współczynniki całkowite. Udowodnić, że jeżeli liczba $ P(5) $ dzieli się przez 2, a liczba $ P(2) $ dzieli się przez 5, to liczba $ P(7) $ dzieli się przez 10.
|
Dowód opiera się na spostrzeżeniu, że jeśli $ P(x) = a_0 + a_1x + \ldots + a_nx^n $ jest wielomianem o współczynnikach całkowitych, to dla każdej pary różnych liczb całkowitych $ u $, $ v $ różnica $ P(u) - P(v) $ jest podzielna przez różnicę $ u-v $. Istotnie:
a każda z różnic $ u^k - v^k $ rozkłada się na czyn... | proof | Algebra | proof | Yes | Yes | olympiads | false |
LIX OM - II - Task 6
Given a positive integer $ n $ not divisible by 3. Prove that there exists a number $ m $ with the following property:
Every integer not less than $ m $ is the sum of the digits of some multiple of the number $ n $. | A good number will be understood as a positive integer that is the sum of the digits of some multiple of the number $ n $.
First of all, note that the sum of two good numbers is also a good number. Indeed, let $ k_1, k_2 $ be multiples of the number $ n $ with the sums of their digits being $ s_1, s_2 $, respectively.... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XVIII OM - III - Task 4
Prove that the polynomial $ x^3 + x + 1 $ is a divisor of the polynomial $ P_n(x) = x^{n + 2} + (x+l)^{2n+1} $ for every integer $ n \geq 0 $. | We apply complete induction. For $ n = 0 $, the theorem is true, since $ P_0(x) = x^2 + x + 1 $. Assume that for some natural number $ n $, the polynomial $ P_n (x) $ is divisible by $ x^2 + x + 1 $. Let's form the polynomial
Hence
From this equality, by the induction hypothesis, it follows that the polynom... | proof | Algebra | proof | Yes | Incomplete | olympiads | false |
XLI OM - II - Problem 6
For any convex polygon $ W $ with area 1, let $ f(W) $ denote the area of the convex polygon whose vertices are the midpoints of all the sides of the polygon $ W $. For each natural number $ n \geq 3 $, determine the lower bound and upper bound of the set of numbers $ f(W) $ as $ W $ runs throu... | The case $ n = 3 $ is trivial: for any triangle $ W $ with area $ 1 $, the equality $ f(W) = 1/4 $ holds; the function $ f $ is constant in this case.
From now on, we assume that $ n \geq 4 $.
Let $ W $ be any $ n $-gon with area $ 1 $. Each of the triangles whose vertices are three consecutive vertices of the polygon... | notfound | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
XXVIII - I - Problem 7
Three circles with radii equal to 1 intersect such that their common part is bounded by three arcs, each belonging to a different circle. The endpoints of these arcs are denoted by $ K, L, M $. The first circle intersects the second at points $ C $ and $ M $, the second intersects the third at p... | If two circles of equal radii intersect at points $ P $ and $ Q $, then the lengths of the arcs determined by these points on one circle are equal to the lengths of the corresponding arcs determined by these points on the other circle. By performing a symmetry with respect to the line $ PQ $, the arcs with endpoints $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXII - I - Problem 3
A cone $ S_1 $ is circumscribed around a sphere $ K $. Let $ A $ be the center of the circle of tangency of the cone with the sphere. A plane $ \Pi $ is drawn through the vertex of the cone $ S_1 $ perpendicular to the axis of the cone. A second cone $ S_2 $ is circumscribed around the sphere $ K... | Let $ O $ be the center of the sphere, $ r $ - the length of its radius. $ W_1 $, $ W_2 $ are the vertices of the cones $ S_1 $ and $ S_2 $ (Fig. 7).
om32_1r_img_7.jpg
The plane determined by the points $ O $, $ W_1 $, $ W_2 $ contains the point $ A $, which is the center of the circle of tangency of the cone $ S_1 $ w... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXVII OM - II - Problem 4
Inside the circle $ S $, place the circle $ T $ and the circles $ K_1, K_2, \ldots, K_n $ that are externally tangent to $ T $ and internally tangent to $ S $, with the circle $ K_1 $ tangent to $ K_2 $, $ K_2 $ tangent to $ K_3 $, and so on. Prove that the points of tangency between the circ... | Let circles $S$ and $T$ be concentric. We will show that the points of tangency $K_1$ with $K_2$, $K_2$ with $K_3$, etc., are equidistant from the common center $O$ of circles $S$ and $T$.
Let for $i = 1, 2, \ldots, A_i$ be the point of tangency of circles $K_i$ and $T$, $B_i$ - the point of tangency of circles $K_i$ a... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LI OM - II - Problem 4
Point $ I $ is the center of the circle inscribed in triangle $ ABC $, where $ AB \neq AC $. Lines $ BI $ and $ CI $ intersect sides $ AC $ and $ AB $ at points $ D $ and $ E $, respectively. Determine all possible measures of angle $ BAC $ for which the equality $ DI = EI $ can hold. | We will show that the only value taken by angle $ BAC $ is $ 60^\circ $.
By the Law of Sines applied to triangles $ ADI $ and $ AEI $, we obtain $ \sin \measuredangle AEI = \sin \measuredangle ADI $. Hence,
om51_2r_img_6.jpg
First, suppose that the equality $ \measuredangle AEI = \measuredangle ADI $ holds (Fig.... | 60 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXXVIII OM - I - Problem 1
The polynomial $ P(x) = a_0+a_1x+ \ldots +a_nx^n $ satisfies the conditions $ P(1) = 0 $ and $ P(x) \geq 0 $ for every $ x \in \mathbb{R} $. Prove that | Polynomial $ P $ has a minimum at the point $ x = 1 $. Therefore, we have the equalities
Multiplying the first of these by $ n+1 $ and subtracting the second, we get | proof | Algebra | proof | Yes | Incomplete | olympiads | false |
XXIV OM - II - Problem 4
Let $ x_n = (p + \sqrt{q})^n - [(p + \sqrt{q})^n] $ for $ n = 1, 2, 3, \ldots $. Prove that if $ p $, $ q $ are natural numbers satisfying the condition $ p - 1 < \sqrt{q} < p $, then $ \lim_{n\to \infty} x_n = 1 $.
Note. The symbol $ [a] $ denotes the greatest integer not greater than $ a $. | We will first prove the
Lemma. If $ p $ and $ q $ are integers, then the number
is an integer for $ n = 0, 1, 2, \ldots $.
Proof. Using the binomial formula, we obtain
If the number $ i $ is even, $ i = 2m $, then $ (\sqrt{q})^i + (-\sqrt{q})^i = 2q^m $. If the number $ i $ is odd, then $ (\sqrt{q})^i + (-\... | proof | Algebra | proof | Yes | Incomplete | olympiads | false |
VIII OM - I - Problem 10
Prove that among all quadrilaterals inscribed in a given circle, the square has the largest area. | Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$ and radius $r$; let the angles $AOB$, $BOC$, $COD$, $DOA$ be denoted by $\alpha$, $\beta$, $\gamma$, $\delta$ respectively. If the center of the circle $O$ lies inside the quadrilateral, then (Fig. 6a)
thus
therefore
with equality hol... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXXIV OM - II - Problem 6
For a given number $ n $, let $ p_n $ denote the probability that when a pair of integers $ k, m $ satisfying the conditions $ 0 \leq k \leq m \leq 2^n $ is chosen at random (each pair is equally likely), the number $ \binom{m}{k} $ is even. Calculate $ \lim_{n\to \infty} p_n $. | om34_2r_img_10.jpg
The diagram in Figure 10 shows Pascal's triangle written modulo $2$, meaning it has zeros and ones in the places where the usual Pascal's triangle has even and odd numbers, respectively. Just like in the usual Pascal's triangle, each element here is the sum of the elements directly above it, accordin... | 1 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
L OM - I - Problem 8
Given a natural number $ n \geq 2 $ and an $ n $-element set $ S $. Determine the smallest natural number $ k $ for which there exist subsets $ A_1, A_2, \ldots, A_k $ of the set $ S $ with the following property:
for any two distinct elements $ a, b \in S $, there exists a number $ j \in \{1, 2, ... | We will show that $ k = [\log_2(n-1) +1] $, i.e., $ 2^{k-1} < n \leq 2^k $.
Given the number $ k $ defined as above, we construct sets $ A_j $ as follows: we number the elements of set $ S $ with $ k $-digit binary numbers (leading zeros are allowed). We thus have $ 2^k \geq n $ numbers available. Then, we take $ A_i $... | [\log_2(n-1)+1] | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XXXIII OM - II - Problem 5
Let $ q $ be a positive even number. Prove that for every natural number $ n $ the number
divides $ (q + 1)^{n+1} $ but does not divide $ (q + 1)^{n+2} $. | We will conduct an inductive proof.
For $ n = 0 $ we have $ q^{(q=1)^0} +1 = q + 1 $. This number is divisible by
$ (q + 1)^{0+1} $, but not by $ (q + 1)^{0+2} $.
Assume that for some $ n \geq 0 $ the conditions are satisfied
which are
We need to show that the conditions will also be satisfied for $ n+1 $
... | proof | Number Theory | proof | Incomplete | Incomplete | olympiads | false |
XLVIII OM - II - Problem 3
Given a set of $ n $ points ($ n \geq 2 $), no three of which are collinear. We color all segments with endpoints in this set such that any two segments sharing a common endpoint have different colors. Determine the smallest number of colors for which such a coloring exists. | Let's assume that the given points are vertices of some $n$-gon (if $n = 2$, of course, one color is enough). All segments of the same color determine disjoint pairs of vertices of the polygon. The maximum number of disjoint two-element subsets of an $n$-element set is $[n/2]$. Therefore, the number of colors is not le... | [n/2] | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XXII OM - II - Problem 3
Given 6 lines in space, of which no 3 are parallel, no 3 pass through the same point, and no 3 are contained in the same plane. Prove that among these 6 lines, there are 3 lines that are mutually skew. | From the conditions of the problem, it follows that in every triple of the considered lines, there is a pair of skew lines. Let us associate the given lines with the vertices of a convex hexagon marked with the numbers $1, 2, \ldots, 6$ (Fig. 11). If two lines are skew, let us connect the corresponding vertices of the ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XIV OM - II - Task 3
Solve the system of equations in integers
Note: The translation is provided as requested, preserving the original text's line breaks and format. However, the system of equations is not provided in the original text. If you have the equations, please provide them, and I can translate the... | The system of equations (1) is equivalent to the system
In the second of the above equations, the left side is a polynomial of degree $3$ in the variables $x$, $y$, $z$, which is identically zero when $x = -y$; this polynomial is therefore divisible by $x + y$. Similarly, we conclude that it is divisible by $y + z$ an... | (1,1,1),(-5,4,4),(4,-5,4),(4,4,-5) | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XII OM - I - Problem 2
Prove that if $ a_1 \leq a_2 \leq a_3 $ and $ b_1 \leq b_2 \leq b_3 $, then | Inequality (1) will be replaced by an equivalent inequality
This, in turn, by the inequality
According to the assumption $ a_1 - a_2 \leq 0 $, $ b_1 -b_2 \leq 0 $, therefore
Hence
Similarly
And
Adding the above 3 inequalities side by side, we obtain inequality (2) equivalent to the desired inequality ... | proof | Inequalities | proof | Yes | Yes | olympiads | false |
LX OM - II - Zadanie 3
Rozłączne okręgi $ o_1 $ i $ o_2 $ o środkach odpowiednio $ I_1 $ i $ I_2 $ są styczne do prostej
$ k $ odpowiednio w punktach $ A_1 $ i $ A_2 $ oraz leżą po tej samej jej stronie. Punkt $ C $ leży na odcinku
$ I_1I_2 $, przy czym $ \meqsuredangle A_1CA_2 =90^{\circ} $. Dla $ i=1,2 $ niech $ B_i... |
Niech prosta styczna do okręgu $ o_1 $ w punkcie $ B_1 $ przecina prostą $ A_1A_2 $ w punkcie $ D $.
(Punkt przecięcia istnieje, gdyż punkt $ C $ nie leży na prostej $ A_1I_1 $, a więc odcinek $ A_1B_1 $
nie jest średnicą okręgu $ o_1 $.) Trójkąt $ A_1B_1D $ jest wówczas równoramienny, gdyż dwa jego boki
są odcinkami ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LV OM - I - Task 4
Given is an acute triangle $ ABC $. We consider all such equilateral triangles $ XYZ $ such that points $ A $, $ B $, $ C $ are internal points of segments $ YZ $, $ ZX $, $ XY $, respectively. Prove that the centroids of all considered triangles $ XYZ $ lie on a single circle. | If triangle $ABC$ is equilateral, then the centroids of all considered triangles $XYZ$ coincide with the centroid of triangle $ABC$. This point, of course, lies on any circle that passes through it.
Assume, therefore, that triangle $ABC$ is not equilateral. Then, without loss of generality, we can assume that $ \measur... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
LVII OM - III - Problem 6
Determine all pairs of integers $ a $, $ b $, for which there exists a polynomial $ P(x) $ with integer coefficients, such that the product $ (x^2 + ax + b)\cdot P(x) $ is a polynomial of the form
where each of the numbers $ c_0,c_1,\dots ,c_{n-1} $ is equal to 1 or -1. | If the quadratic polynomial $x^2 + ax + b$ is a divisor of some polynomial $Q(x)$ with integer coefficients, then of course the constant term $b$ is a divisor of the constant term of the polynomial $Q(x)$. Therefore, in the considered situation, $b=\pm 1$.
No polynomial $Q(x)$ of the form $x^n \pm x^{n-1} \pm x^{n-2} \... | (-2,1),(-1,-1),(-1,1),(0,-1),(0,1),(1,-1),(1,1),(2,1) | Algebra | math-word-problem | Yes | Yes | olympiads | false |
III OM - I - Problem 6
Prove that if $ u \geq 0 $, $ v \geq 0 $, $ w \geq 0 $, then the inequality holds
The translation is provided as requested, maintaining the original text's line breaks and format. | When one of the numbers $ u $, $ v $, $ w $ equals zero, the inequality above is true, since the right side then equals zero, while the left side is non-negative. Therefore, we only need to prove that when $ u > 0 $, $ v > 0 $, $ w > 0 $, then
\spos{I} We will factor the polynomial $ u^3 + v^3 + w^3 - 3uvw $. For... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
II OM - III - Problem 5
In a circle, a quadrilateral $ABCD$ is inscribed. Lines $AB$ and $CD$ intersect at point $E$, and lines $AD$ and $BC$ intersect at point $F$. The angle bisector of $\angle AEC$ intersects side $BC$ at point $M$ and side $AD$ at point $N$; and the angle bisector of $\angle BFD$ intersects side $... | Ignoring for now the condition that points $A$, $B$, $C$, $D$ lie on a circle, consider any convex quadrilateral $ABCD$ where the extensions of sides $AB$ and $CD$ intersect at point $E$, and the extensions of sides $AD$ and $BC$ intersect at point $F$.
Draw the angle bisectors $EO$ and $FO$ of angles $E$ and $F$, and ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXII - III - Problem 6
In a tetrahedron with volume $ V $, the sum of the squares of the lengths of all edges is equal to $ S $. Prove that
| Consider the tetrahedron $ABCD$ (Fig. 15), let point $O$ be the foot of the perpendicular dropped from the tetrahedron to the base $ABC$, and let $DO = H$. Let $E$ be the foot of the perpendicular dropped from the triangle $ABC$ to the base $\overline{AB}$, and let $CE = h$. Let $F$ be the foot of the perpendicular dro... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XXX OM - II - Task 1
Given are points $ A $ and $ B $ on the edge of a circular pool. An athlete needs to get from point $ A $ to point $ B $ by walking along the edge of the pool or swimming in the pool; he can change his mode of movement multiple times. How should the athlete move to get from point $ A $ to $ B $ in... | Let the radius of the pool be $r$, and the arc $\widehat{AB}$ corresponds to the central angle $\alpha$, where $0 < \alpha \leq \pi$. Then the length of the arc $\widehat{AB}$ is $r\alpha$, and the length of the segment $\overline{AB}$ is (Fig. 10)
om30_2r_img_10.jpg
We will prove that $r\alpha < 2AB$. It will follow ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
LIX OM - III - Task 3
In a convex pentagon $ABCDE$, where $BC = DE$, the following equalities hold:
Prove that the quadrilateral $BCDE$ is a parallelogram. | First of all, let us note (Fig. 1) that if point \( I \) is the center of the circle inscribed in triangle \( XYZ \), then the following equality holds:
\[
\]
For the proof, it is sufficient to see that
\[
\]
Therefore,
\[
\]
We proceed to the solution of the problem. Let us denote
\[
\]
Let \( o \) be the circ... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XXXI - II - Task 2
Prove that for any real numbers $ x_1, x_2, x_3, \ldots, x_n $ the inequality is true
(Note: The original text did not specify the inequality to be proven. If you have the specific inequality, please provide it for a complete translation.) | Let's first assume that $ x_1, x_2, \ldots, x_n $ are non-negative numbers. Consider the numbers
the geometric mean of these numbers is
the arithmetic mean is
By the inequality between the arithmetic mean and the geometric mean, we obtain
If the numbers $ x_1, x_2, \ldots, x_n $ are not all non-negative, then by th... | proof | Inequalities | proof | Incomplete | Incomplete | olympiads | false |
XXVIII - I - Task 3
Let $ a $ and $ b $ be natural numbers. A rectangle with sides of length $ a $ and $ b $ has been divided by lines parallel to the sides into unit squares. Through the interiors of how many squares does the diagonal of the rectangle pass? | Consider a rectangle $ABCD$ with base $\overline{AB}$ of length $a$ and height $\overline{BC}$ of length $b$. Let's consider its diagonal $\overline{AC}$ connecting the lower left vertex $A$ with the upper right vertex $C$ (Fig. 4).
First, assume that the numbers $a$ and $b$ are relatively prime. If a vertex $E$ of a u... | \gcd(,b) | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXVII OM - III - Problem 2
Given are four sequences of real numbers $ (a_n) $, $ (b_n) $, $ (c_n) $, $ (d_n) $, such that for
each $ n $
Prove that if for some $ k, m \geq 1 $ it is $ a_{k+m} = a_m $, $ b_{k+m} = b_m $, $ c_{k+m} = c_m $, then $ a_2=b_2=c_2=d_2 $. | From the conditions of the problem, it follows that the sequences $ (a_n) $, $ (b_n) $, $ (c_n) $, $ (d_n) $ are periodic from some point onward, and thus the sequences $ A_n = a_n + b_n + c_n + d_n $ and $ B_n = a_n^2 + b_n^2 + c_n^2 + d_n^2 $ are also periodic from some point onward. We also have $ A_{n+1} = a_{n+1} ... | a_2=b_2=c_2=d_2=0 | Algebra | proof | Incomplete | Yes | olympiads | false |
XLV OM - III - Problem 6
Different real numbers $ x_1, x_2, \ldots , x_n $ ($ n \geq 4 $) satisfy the conditions
Prove that among these numbers, one can choose four different numbers $ a $, $ b $, $ c $, $ d $, such that the inequalities hold
(Note: The problem statement seems to be missing the spec... | Let us choose four different numbers $a$, $b$, $c$, $d$ from the set $\{x_1, x_2, \ldots, x_n\}$ in any way and consider the polynomials
and
From the conditions given in the assumptions of the problem, the following equalities follow
which means
and similarly,
The problem will be solved if we can choose ... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
XLVI OM - II - Problem 5
The circles inscribed in the faces $ABC$ and $ABD$ of the tetrahedron $ABCD$ are tangent to the edge $AB$ at the same point. Prove that the points of tangency of these circles with the edges $AC$, $BC$ and $AD$, $BD$ lie on a single circle. | Let's assume that the circle inscribed in the face $ABC$ is tangent to the edges $AC$, $BC$, and $AB$ at points $P$, $Q$, and $T$, respectively, and the circle inscribed in the face $ABD$ is tangent to the edges $AD$, $BD$, and $AB$ at points $R$, $S$, and $T$ (the same point $T$, according to the assumption). We have ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LX OM - II - Task 1
Real numbers $ a_1, a_2, \cdots , a_n $ $ (n \geqslant 2) $ satisfy the condition
$ a_1 \geqslant a_2 \geqslant \cdots \geqslant a_n > 0 $. Prove the inequality | We will apply mathematical induction with respect to the value of the number $ n $.
For $ n = 2 $, the thesis of the problem is true, as both sides of the inequality are equal to $ 2a_2 $.
Moving on to the inductive step, consider the real numbers
$ a_1, a_2, \cdots, a_{n+1} $, for which $ a_1 \geqslant a_2 \geqslant... | proof | Inequalities | proof | Yes | Yes | olympiads | false |
XVIII OM - II - Task 6
Prove that points $ A_1, A_2, \ldots, A_n $ ($ n \geq 7 $) lying on the surface of a sphere lie on a circle if and only if the tangent planes to the sphere at these points have a common point or are parallel to a single line. | a) Suppose that points $ A_1, A_2, \ldots, A_n $ on the surface of a sphere with center $ O $ lie on a circle $ k $. Let $ \alpha_1 $ denote the tangent plane to the sphere at point $ A_i $ ($ i= 1,2, \ldots, n $), and $ \pi $ - the plane of the circle $ k $.
If the center of the circle $ k $ is point $ O $, then the p... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XVI OM - I - Problem 4
The school organized three trips for its 300 students. The same number of students participated in each trip. Each student went on at least one trip, but half of the participants in the first trip, one-third of the participants in the second trip, and one-fourth of the participants in the third ... | Let $ x $ denote the number of participants in each trip, and $ y $, $ z $, $ u $, $ w $ denote, respectively, the numbers of students who went: a) only on the first and second trip, b) only on the first and third trip, c) only on the second and third trip, d) on all three trips.
The total number of students equals the... | x=120,y=14,z=27,u=53,w=37 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XI OM - I - Problem 11
Prove that if a quadrilateral is circumscribed around a circle, then a circle can be circumscribed around it if and only if the chords connecting the points of tangency of the opposite sides of the quadrilateral with the circle are perpendicular. | Let $ABCD$ be a quadrilateral inscribed in a certain circle, and let $M$, $N$, $P$, $Q$ denote the points of tangency of the sides $AB$, $BC$, $CD$, $DA$ with the inscribed circle, and let $S$ be the point of intersection of segments $MP$ and $NQ$ (Fig. 16).
Since tangents to a circle drawn from the ends of a chord for... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LIV OM - III - Task 3
Determine all polynomials $ W $ with integer coefficients that satisfy the following condition: for every natural number $ n $, the number $ 2^n-1 $ is divisible by $ W(n) $. | Let $ W(x) = a_0 + a_1x +\ldots + a_rx^r $ be a polynomial with integer coefficients, satisfying the given condition. Suppose that for some natural number $ k $, the value $ W(k) $ is different from $ 1 $ and $ -1 $; it therefore has a prime divisor $ p $. Let $ m = k + p $. The difference $ W(m) - W(k) $ is the sum of... | W(x)=1orW(x)=-1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
LII OM - I - Problem 8
Determine for which pairs of natural numbers $ m $ and $ n $ a rectangle with side lengths $ m $ and $ n $ can be cut into parts congruent to the figure in the diagram. Each of the four squares in the diagram has a side length of $ 1 $.
om52_1r_img_10.jpg | Answer: The desired division is possible if and only if $ m,n > 1 $ and the product $ mn $ is divisible by $ 8 $.
First, assume that we managed to cut the rectangle $ m \times n $ into parts congruent to the given figure (which we will call blocks in the rest of the solution). Of course, $ m,n > 1 $. We will show that ... | Thedesireddivisionispossibleifonlyif\(,n>1\)theproduct\(mn\)isdivisible\(8\) | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
XXXVII OM - II - Problem 5
Prove that if a polynomial $ f $ not identically zero satisfies for every real $ x $ the equation $ f(x)f(x + 3) = f(x^2 + x + 3) $, then it has no real roots. | Suppose a polynomial $ f $ satisfying the given conditions has real roots. Let $ x_0 $ be the largest of them. The number $ y_0 = x_0^2 + x_0 + 3 > x_0 $ is also a root of the polynomial $ f $, because
- contrary to the fact that $ x_0 $ is the largest root. The obtained contradiction proves that $ f $ cannot have re... | proof | Algebra | proof | Yes | Yes | olympiads | false |
VI OM - II - Task 3
What should be the angle at the vertex of an isosceles triangle so that a triangle can be constructed with sides equal to the height, base, and one of the remaining sides of this isosceles triangle? | We will adopt the notations indicated in Fig. 9. A triangle with sides equal to $a$, $c$, $h$ can be constructed if and only if the following inequalities are satisfied:
Since in triangle $ADC$ we have $a > h$, $\frac{c}{2} + h > a$, the first two of the above inequalities always hold, so the necessary and suffic... | 106 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XIX OM - I - Problem 9
Prove that if $ a_1, a_2, \ldots, a_n $ denote the numbers $ 1, 2, \ldots, n $ arranged in any order, and $ n $ is an odd number, then the number $ N = (a_1 - 1)(a_2 - 2) \ldots (a_n - n) $ is even. | If the product $ N $ were an odd number, then each of its $ n $ factors $ a_1 - 1, a_2 - 2, \ldots, a_n - n $ would also be an odd number. For an odd $ n $, the sum of these factors would be an odd number, which is not true, since | proof | Number Theory | proof | Yes | Incomplete | olympiads | false |
XXXIV OM - I - Problem 5
Prove the existence of a number $ C_0 $ with the property that for any sequence $ x_1, x_2, \ldots, x_N $ of positive numbers and for any positive number $ K $, if the number of terms $ x_j $ not less than $ K $ is not greater than $ \frac{N}{K} $, then | Suppose the sequence $ x_1, x_2, \ldots, x_N $ satisfies the given condition. By rearranging the terms if necessary, we can assume that $ x_1 \geq x_2 \geq \ldots \geq x_N $. We will show that $ x_j \leq N/j $ for $ j = 1, \ldots, N $. Suppose this is not the case, i.e., for some index $ m $, the inequality $ x_m > N/m... | proof | Inequalities | proof | Yes | Yes | olympiads | false |
XXXVI OM - III - Problem 2
Given a square with side length 1 and positive numbers $ a_1, b_1, a_2, b_2, \ldots, a_n, b_n $ not greater than 1 such that $ a_1b_1 + a_2b_2 + \ldots + a_nb_n > 100 $. Prove that the square can be covered by rectangles $ P_i $ ($ i = 1,2,\ldots,n $) with side lengths $ a_i, b_i $ parallel ... | Let $ AB $ be the base of a square. We can assume that $ 1 \geq a_x \geq a_2 \geq \ldots \geq a_n $. On the ray $ AB^\to $, we sequentially lay down segments of lengths $ b_1, b_2, \ldots, b_k $ until we exceed point $ B $, and over each subsequent segment of length $ b_i $, we construct a rectangle whose other side ha... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XVIII OM - I - Problem 2
Prove that
for all values of $ a $ for which the left side of this inequality makes numerical sense. | The left side of inequality (1) makes numerical sense if and only if $ - \frac{1}{3} \leq a \leq \frac{17}{5} $; let's assume that the number $ a $ satisfies this condition.
Notice that the sum of the squares of the terms on the left side of (1) equals $ 21 $; the problem thus reduces to finding the maximum of the sum ... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
XVII OM - I - Problem 8
In space, three lines are given. Find a line that intersects them at points $ A, B, C $ such that $ \frac{AB}{BC} $ equals a given positive number $ k $. | Let us first consider the case where the three given lines are pairwise skew, and at the same time do not lie in three mutually parallel planes.
Let the given lines be denoted by the letters $a$, $b$, $c$ and let $l$ be the line intersecting them at points $A$, $B$, $C$ respectively, such that $\frac{AB}{BC}=k$.
Let $\... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XLV OM - I - Problem 8
Given natural numbers $ a $, $ b $, $ c $, such that $ a^3 $ is divisible by $ b $, $ b^3 $ is divisible by $ c $, and $ c^3 $ is divisible by $ a $. Prove that the number $ (a+b+c)^{13} $ is divisible by $ abc $. | The number $ (a + b + c)^{13} $ is the product of thirteen factors, each equal to $ a + b + c $. Multiplying these thirteen identical expressions, we obtain a sum of many (specifically $ 3^{13} $) terms of the form
It suffices to show that each such term is divisible by the product $ abc $. This is obvious when none o... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
I OM - B - Problem 15
Two intersecting circles with radii $ r $ and $ R $ are given. Through one of the intersection points of these circles, draw a line such that this line intersects the circles at two more points $ p $ and $ Q $ and the segment $ PQ $ has a given length $ d $.
om1_Br_img_8.jpg | Through the intersection point $ A $ of the given circles with centers $ O $ and $ S $, we draw a line intersecting these circles at points $ P $ and $ Q $.
om1_Br_img_9.jpg
Points $ P $ and $ Q $ lie either on opposite sides of point $ A $ (Fig. 8), or on the same side of point $ A $ (Fig. 9).
In the first case, $ PQ ... | notfound | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
XLVII OM - III - Problem 3
Given a natural number $ n \geq 2 $ and positive numbers $ a_1, a_2, \ldots , a_n $, whose sum equals 1.
(a) Prove that for any positive numbers $ x_1, x_2, \ldots , x_n $ whose sum equals 1, the following inequality holds:
(b) Determine all systems of positive numbers $ x_1, x_2, \ldots , ... | From the given conditions ($ n \geq 2 $; $ a_i>0 $; $ \sum a_i =1 $), it follows that the numbers $ a_1, a_2, \ldots, a_n $ belong to the interval $ (0;\ 1) $. Let us assume
these are also numbers in the interval $ (0;\ 1) $. For them, the equality
holds.
Let $ x_1, x_2,\ldots, x_n $ be any real numbers whose sum is... | proof | Inequalities | proof | Yes | Yes | olympiads | false |
XXIX OM - I - Problem 1
The sequence of numbers $ (p_n) $ is defined as follows: $ p_1 = 2 $, $ p_n $ is the largest prime divisor of the sum $ p_1p_2\ldots p_{n-1} + 1 $. Prove that the number 5 does not appear in the sequence $ (p_n) $. | We have $ p_1 = 2 $ and $ p_2 = 3 $, because $ p_2 $ is the largest prime divisor of the number $ p_1 + 1 = 3 $. Therefore, for $ n > 2 $, the number $ p_1p_2 \ldots p_{n-1} + 1 = 6p_3 \ldots p_n + 1 $ is not divisible by either $ 2 $ or $ 3 $. If, for some natural number $ n $, the largest prime divisor of the number ... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
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